Difference between revisions of "Simon's Favorite Factoring Trick"
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| − | + | Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user ComplexZeta, whose name is Simon. [http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=factoring&t=8215 This] appears to be the thread where '''Simon's favorite factoring trick''' was first introduced. | |
| − | Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization. | ||
| − | === | + | == Statement of the factorization == |
| + | The general statement of SFFT is: <math>{xy}+{xk}+{yj}+{jk}=(x+j)(y+k)</math>. More oftenly however SFFT is introduced as <math>xy + x + y + 1 = (x+1)(y+1)</math> or <math> xy - x - y +1 = (x-1)(y-1)</math>. | ||
| − | + | == Applications == | |
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This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>{x}</math> and <math>{y}</math> are variables and <math>j,k</math> are known constants. Also it is typically necessary to add the <math>{j}{k}</math> term to both sides to perform the factorization. | This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>{x}</math> and <math>{y}</math> are variables and <math>j,k</math> are known constants. Also it is typically necessary to add the <math>{j}{k}</math> term to both sides to perform the factorization. | ||
| − | + | == Examples == | |
([[AIME]] 1987/5) <math>m</math> and <math>n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>. | ([[AIME]] 1987/5) <math>m</math> and <math>n</math> are integers such that <math>m^2 + 3m^2n^2 = 30n^2 + 517</math>. Find <math>3m^2n^2</math>. | ||
| − | '''Outline Solution:''' Rearrange to <math>m^2 + 3m^2n^2 -30n^2= 517</math>. The key step is changing the equation to <math>m^2 + 3m^2n^2 -30n^2-10= 507</math>, where the equation factors to <math>(3n^2 + 1)(m^2 - 10) = 507 = 3\cdot 13^2</math> | + | '''Outline Solution:''' Rearrange to <math>m^2 + 3m^2n^2 -30n^2= 517</math>. The key step is changing the equation to <math>m^2 + 3m^2n^2 -30n^2-10= 507</math>, where the equation factors to <math>(3n^2 + 1)(m^2 - 10) = 507 = 3\cdot 13^2</math>. This makes things much simpler. The rest of the problem is left as an exercise to the reader. |
Revision as of 11:39, 20 June 2006
Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user ComplexZeta, whose name is Simon. This appears to be the thread where Simon's favorite factoring trick was first introduced.
Statement of the factorization
The general statement of SFFT is: 
. More oftenly however SFFT is introduced as 
or 
.
Applications
This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually 
and 
are variables and 
are known constants. Also it is typically necessary to add the 
term to both sides to perform the factorization.
Examples
(AIME 1987/5) 
and 
are integers such that 
. Find 
.
Outline Solution: Rearrange to 
. The key step is changing the equation to 
, where the equation factors to 
. This makes things much simpler. The rest of the problem is left as an exercise to the reader.
